Comptes Rendus
Partial Differential Equations/Mathematical Physics
A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary
[Fluide visqueux dans un domaine de faible épaisseur vérifiant la condition de glissement sur une frontière lègèrement rugueuse]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 967-971.

On considère un fluide visqueux de faible épaisseur ε sur un fond rugueux Γε, périodique de période rε et amplitude δε, δεrεε, où on impose la condition de glissement. Quand ε converge vers zéro on obtient un système de type Reynolds qui dépend de la limite λ de (δεε)/(rεrε). Si λ=+, le fluide se comporte comme si on aurait imposé la condition d'adhérence sur Γε. Ceci justifie la condition usuelle pour un fluide visqueux. Si λ=0 le fluide se comporte comme si Γε était plate. Enfin, pour λ(0,+), tout se passe comme si Γε était plate, mais avec un coefficient de frottement plus élevé.

We consider a viscous fluid of small height ε on a periodic rough bottom Γε of period rε and amplitude δε, δεrεε, where we impose the slip boundary condition. When ε tends to zero we obtain a Reynolds system depending on the limit λ of (δεε)/(rεrε). If λ=+, the fluid behaves as if we would impose the adherence condition on Γε. This justifies why this is the usual boundary condition for viscous fluids. If λ=0 the fluid behaves as if Γε was plane. Finally, for λ(0,+) it behaves as if Γε was flat but with a higher friction coefficient.

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DOI : 10.1016/j.crma.2010.07.023

Juan Casado-Díaz 1 ; Manuel Luna-Laynez 1 ; Francisco Javier Suárez-Grau 1

1 Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain
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     title = {A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary},
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Juan Casado-Díaz; Manuel Luna-Laynez; Francisco Javier Suárez-Grau. A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 967-971. doi : 10.1016/j.crma.2010.07.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.023/

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